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A Unified Explanation of the Kadowaki--Woods Ratio in Strongly Correlated Metals

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A Unified Explanation of the Kadowaki--Woods Ratio in Strongly Correlated Metals LETTERS PUBLISHED ONLINE: 19 APRIL 2009 | DOI: 10.1038/NPHYS1249 A unified explanation of the Kadowaki–Woods ratio in strongly correlated metals A. C. Jacko1, J. O. Fjærestad2 and B. J. Powell1* Discoveries of ratios whose values are constant within broad 104 K-NCS classes of materials have led to many deep physical insights. K-Br The Kadowaki–Woods ratio (KWR; refs 1, 2) compares the 103 UBe13 temperature dependence of a metal’s resistivity to that of its 2 10 CeCu heat capacity, thereby probing the relationship between the 6 1 CeCu2Si2 electron–electron scattering rate and the renormalization of 10 ⊥ Sr2RuO4 LiV2O4 the electron mass. However, the KWR takes very different ) 0 10 Na0.7CoO2 UPt3 3,4 ¬2 values in different materials . Here we introduce a ratio, CeB6 ¬1 β-I closely related to the KWR, that includes the effects of 10 3 β-IBr UAl2 cm K 2 Tl2Ba2CuO6 carrier density and spatial dimensionality and takes the same μΩ ¬2 Rb3C60 ( 10 (predicted) value in organic charge-transfer salts, transition- A ⎢⎢ Sr2RuO4 metal oxides, heavy fermions and transition metals—despite 10¬3 La1.7Sr0.3CuO4 the numerator and denominator varying by ten orders of ¬4 Heavy fermions 10 Pd Organics magnitude. Hence, in these materials, the same emergent Fe Transition metals physics is responsible for the mass enhancement and the 10¬5 Re Ni Pt Oxides quadratic temperature dependence of the resistivity, and no a 10¬6 Os TM exotic explanations of their KWRs are required. aHF In a Fermi liquid the electronic contribution to the heat capacity 100 101 102 103 104 105 106 γ has a linear temperature dependence, that is, Cel(T) D T. Another γ 2 (mJ2 mol¬2 K¬4) prediction of Fermi-liquid theory5 is that, at low temperatures, 2 the resistivity varies as ρ(T) D ρ0 C AT . This is observed Figure 1 j The standard Kadowaki–Woods plot. It can be seen that the experimentally when electron–electron scattering, which gives rise data for the transition metals and heavy fermions (other than UBe13) to the quadratic term, dominates over electron–phonon scattering. fall onto two separate lines. However, a wide range of other strongly 1 2 In a number of transition metals A/γ ≈ aTM D 0:4 µ cm correlated metals do not fall on either line or between the two lines. 2 2 −2 2 2 2 −2 mol K J (Fig. 1), even though γ varies by an order of magnitude aTM D 0:4 µ cm mol K J is the value of the KWR observed in the 2 1 2 2 −2 across the materials studied. Later, it was found that in many transition metals and aHF D 10 µ cm mol K J is the value seen 2 2 2 −2 2 heavy-fermion compounds A/γ ≈ aHF D 10 µ cm mol K J in the heavy fermions . In labelling the data points we use the following 2 (Fig. 1), despite the large mass renormalization, which causes γ abbreviations: κ-Br is κ-(BEDT-TTF)2CuTN(CN)2UBr; κ-NCS is to vary by more than two orders of magnitude in these materials. κ-(BEDT-TTF)2Cu(NCS)2; β-I3 is β-(BEDT-TTF)2I3; and β-IBr2 is 2 Because of this remarkable behaviour A/γ has become known as β-(BEDT-TTF)2IBr2. For Sr2RuO4 we show data for A measured with the the Kadowaki–Woods ratio. However, it has long been known2,6 current both perpendicular and parallel to the basal plane; these data that the heavy-fermion material UBe13 has an anomalously large points are distinguished by the symbols ? and k respectively. Further KWR. More recently, studies of other strongly correlated metals, details of the data are reported in Supplementary Information. such as the transition-metal oxides3,7 and the organic charge- transfer salts4,8, have found surprisingly large KWRs (Fig. 1). It There have been a number of studies of the KWR based on is therefore clear that the KWR is not the same in all metals; in specific microscopic Hamiltonians (see, for example, refs 9–13). fact, it varies by more than seven orders of magnitude across the However, if the KWR has something general to tell us about materials shown in Fig. 1. strongly correlated metals, then we would also like to under- Several important questions need to be answered about the stand which features of the ratio transcend specific microscopic KWR. (1) Why is the ratio approximately constant within the models. Nevertheless, two important points emerge from these transition metals and within the heavy fermions (even though microscopic treatments of the KWR: (1) if the momentum dep- many-body effects cause large variations in their effective masses)? endence of the self-energy can be ignored then the many-body (2) Why is the KWR larger for the heavy fermions than it is renormalization effects on A and γ 2 cancel, and (2) material- for the transition metals? (3) Why are such large and varied specific parameters are required to reproduce the experimentally KWRs observed in layered metals such as the organic charge- observed values of the KWR. Below we investigate the KWR transfer salts and transition-metal oxides? The main aim of using a phenomenological Fermi-liquid theory; this work builds this paper is to resolve question (3). We shall also make on previous studies of related models3,6. Indeed, our calculation some comments on the first two questions, which have been is closely related to that in ref. 6. The main results reported extensively studied previously. here are the identification of a ratio (equation (5)) relating A 1Centre for Organic Photonics and Electronics, School of Physical Sciences, University of Queensland, Brisbane, Queensland 4072, Australia, 2School of Physical Sciences, University of Queensland, Brisbane, Queensland 4072, Australia. *e-mail: [email protected]. 422 NATURE PHYSICS j VOL 5 j JUNE 2009 j www.nature.com/naturephysics © 2009 Macmillan Publishers Limited. All rights reserved. NATURE PHYSICS DOI: 10.1038/NPHYS1249 LETTERS γ 2 and , which we predict takes a single value in a broad class 104 Transition metals K-NCS of strongly correlated metals, and the demonstration that this Heavy fermions 103 Organics K-Br ratio does indeed describe the data for a wide variety of strongly UBe13 correlated metals (Fig. 2). Oxides 102 It has been argued that the KWR is larger in the heavy CeCu6 fermions than the transition metals because the former are 1 10 CeCu2Si2 Sr RuO ⊥ more strongly correlated (in the sense that the self-energy is LiV2O4 2 4 6 ) UPt more strongly frequency dependent) than the latter . Several ¬2 100 Na0.7CoO2 3 β-I scenarios have been proposed to account for the large KWRs 3 CeB6 cm K 10¬1 UAl2 β observed in UBe13, transition-metal oxides and organic charge- -IBr2 6 μΩ transfer salts, including impurity scattering , proximity to a ( Tl2Ba2CuO6 A ¬2 7 10 Sr RuO ⎢⎢ quantum critical point and the suggestion that electron–phonon 2 6 Rb3C60 scattering in reduced dimensions might give rise to a quadratic 10¬3 La Sr CuO temperature dependence of the resistivity14. It has been previously 1.7 0.3 4 3 observed that using volumetric (rather than molar) units for 10¬4 Pd γ reduces the variation in the KWRs of the transition-metal Ni oxides. However, even in these units, the organic charge-transfer 10¬5 Pt Fe Re salts have KWRs orders of magnitude larger than those of other Os strongly correlated metals. We shall argue that the different 10¬132 10¬130 10¬128 10¬126 10¬124 10¬122 γ 2 4 9 ¬6 ¬4 KWRs observed across this wide range of materials result from /fdx (n) (kg m s K ) the simple fact that the KWR contains a number of material- specific quantities. As a consequence, when we replace the KWR Figure 2 j Comparison of the ratio defined in equation (5) with with a ratio that accounts for these material-specific effects experimental data. It can be seen that, in all of the materials studied, (equation (5)) the data for all of these materials do indeed lie on the data are in excellent agreement with our prediction (line). The a single line (Fig. 2). abbreviations in the data-point labels are the same as in Fig. 1. Further Many properties of strongly correlated Fermi liquids can be details of the data are given in Supplementary Information. understood in terms of a momentum-independent self-energy15,16. Therefore, following ref. 6, we assume that the imaginary part of the (DOS) at the Fermi energy. In the low-temperature, pure limit we self-energy, Σ 00(!;T), at energy !, is given by find (see the Methods section) that 2 !2 . /2 16nkB 00 ~ C πkBT A D (3) Σ (!;T) D − −s (1) 2h 2 i 2!∗2 ∗2 π~e v0x D0 2τ0 ! where h···i denotes an average over the Fermi surface. Note that 2 2 ∗2 for j! C(πkBT) j < ! and neither the DOS nor the Fermi velocity are renormalized in this expression. Indeed, all of the many-body effects are encapsulated ∗ 00 = ∗ by ! , which determines the magnitude of the frequency-dependent Σ (!;T) D −[~=2τ CsUF(T!2 C(πk T)2U1 2=! ) 0 B term in Σ 00(!;T); see equation (1). The Kramers–Kronig relation for the retarded self-energy19,20 2 2 ∗2 for j! C (πkBT) j > ! , where 2s=~ is the scattering rate can be used to show (see the Methods section) that, in the due to electron–electron scattering in the absence of quantum pure limit, τ −1 many-body effects, 0 is the impurity scattering rate, F is a monotonically decreasing function with boundary conditions @Σ 0 4s ξ F(1) D 1 and F(1) D 0 and !∗ is determined by the strength of γ D γ 1− D γ 1C u 0 @! 0 π!∗ the many-body correlations.
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